Properties

Label 2-9200-1.1-c1-0-77
Degree $2$
Conductor $9200$
Sign $-1$
Analytic cond. $73.4623$
Root an. cond. $8.57101$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·3-s − 3.32·7-s + 0.0262·9-s − 5.77·11-s + 1.10·13-s + 0.893·17-s − 2.42·19-s + 5.77·21-s − 23-s + 5.17·27-s + 4.11·29-s + 9.54·31-s + 10.0·33-s − 7.69·37-s − 1.91·39-s + 0.00418·41-s + 9.97·43-s − 10.0·47-s + 4.03·49-s − 1.55·51-s + 6.25·53-s + 4.22·57-s + 10.7·59-s + 10.5·61-s − 0.0871·63-s − 10.9·67-s + 1.73·69-s + ⋯
L(s)  = 1  − 1.00·3-s − 1.25·7-s + 0.00874·9-s − 1.74·11-s + 0.305·13-s + 0.216·17-s − 0.557·19-s + 1.26·21-s − 0.208·23-s + 0.995·27-s + 0.763·29-s + 1.71·31-s + 1.74·33-s − 1.26·37-s − 0.306·39-s + 0.000653·41-s + 1.52·43-s − 1.45·47-s + 0.576·49-s − 0.217·51-s + 0.858·53-s + 0.559·57-s + 1.40·59-s + 1.35·61-s − 0.0109·63-s − 1.33·67-s + 0.209·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9200\)    =    \(2^{4} \cdot 5^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(73.4623\)
Root analytic conductor: \(8.57101\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{9200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 + T \)
good3 \( 1 + 1.73T + 3T^{2} \)
7 \( 1 + 3.32T + 7T^{2} \)
11 \( 1 + 5.77T + 11T^{2} \)
13 \( 1 - 1.10T + 13T^{2} \)
17 \( 1 - 0.893T + 17T^{2} \)
19 \( 1 + 2.42T + 19T^{2} \)
29 \( 1 - 4.11T + 29T^{2} \)
31 \( 1 - 9.54T + 31T^{2} \)
37 \( 1 + 7.69T + 37T^{2} \)
41 \( 1 - 0.00418T + 41T^{2} \)
43 \( 1 - 9.97T + 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 - 6.25T + 53T^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 - 1.89T + 73T^{2} \)
79 \( 1 - 0.216T + 79T^{2} \)
83 \( 1 - 5.38T + 83T^{2} \)
89 \( 1 - 6.00T + 89T^{2} \)
97 \( 1 - 2.08T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.16048649115575449025534754660, −6.54800845864513703823999376509, −6.04773173920008072507908718025, −5.36227174569055420954161187839, −4.84466678345085658432484078497, −3.84322512381899499998837327537, −2.93748717931322699017654077874, −2.40368608223073436649612338913, −0.821671778994400008396829252709, 0, 0.821671778994400008396829252709, 2.40368608223073436649612338913, 2.93748717931322699017654077874, 3.84322512381899499998837327537, 4.84466678345085658432484078497, 5.36227174569055420954161187839, 6.04773173920008072507908718025, 6.54800845864513703823999376509, 7.16048649115575449025534754660

Graph of the $Z$-function along the critical line